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RANDOM WALKS ON BRATTELI DIAGRAMS
Jean Renault
To cite this version:
Jean Renault. RANDOM WALKS ON BRATTELI DIAGRAMS. 2017. �hal-01514270�
JEAN RENAULT
Abstract. In a 1989 article, A. Connes and E. J. Woods made a connection between hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks. I will explain this connection and will present two theorems given there: the description of an almost periodic state on a hyperfinite von Neumann algebra (due to A.
Connes) and the ergodic decomposition of a Markov measure via harmonic functions (a classical result in J. Neveu 64). The crux of the first theorem is a model for conditional expectations on finite dimensional C*-algebras. Our proof of the second theorem hinges on the notion of cotransition probability.
1. Introduction.
This article is based on
A. Connes and E. J. Woods , Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks, Pacific J. Math. 137 (1989), no 2, 225-243.
It contains the statement of the two theorems which I am going to describe:
(i) the description of an arbitrary state on a hyperfinite von Neumann algebra (due to A. Connes);
(ii) the ergodic decomposition of a Markov measure via harmonic functions (a classical result in J. Neveu 64).
2. Conditional expectations on finite dimensional C*-algebras Although the material of this section is not new, I have not found a reference for Theorem 2.8 below which gives a complete invariant for a faithful conditional expectation on a finite dimensional C*-algebra. The description of an inclusion of finite dimensional C*-algebras in terms of matrix units, which is a part of the theorem, and its graphical description, have been given by O. Bratteli in his fundamental paper [3] (see Proposition 1.7 and section 1.8). The graphical description of a conditional expectation on a finite dimensional C*-algebra appears in section 3 (iii) of [8] but without much detail. The only originality may be our systematic use of Cartan subalgebras. The corresponding groupoid models are also known as path models or tail equivalence relations.
We first give the ingredients and the recipe to construct a faithful conditional expec- tation Qof a finite dimensional C*-algebraM onto a sub-C*-algebra M. Then, we show that every faithful conditional expectation is obtained by this construction.
We first recall that we can associate to an equivalence relation R on a finite set X a finite dimensional C*-algebra M = C∗(R): its elements are the functions f : R → C
1991Mathematics Subject Classification. Primary 22A22; Secondary 54H20, 43A65, 46L55.
Key words and phrases. Bratteli diagrams, hyperfinite von Neumann algebras.
1
(here R ⊂ X ×X is the graph of the equivalence relation), the product is the matrix multiplication and the involution is the usual complex conjugate of a matrix. It has a canonical matrix unit (e(x, y))(x,y)∈R indexed by R.
We consider now finite sets X, V, E, V equipped with surjections r :X →V, s :E →V, r:E →V. We define
X ={(x, a)∈X×E :r(x) = s(a)}
and the equivalence relations:
R ={(x, y)∈X×X :r(x) = r(y)}
R ={(xa, yb)∈X×X :r(a) =r(b).
We construct the C*-algebras C∗(R) andC∗(R). The mapj :C∗(R)→C∗(R) given by j(f)(xa, yb) =
f(x, y) if a=b 0 if a6=b
identifiesC∗(R) to a subalgebra of C∗(R). We shall make this identification and view the elements of C∗(R) as functions on R. Then (V, E, V) is the graph of the inclusion. We leave as an exercise to the reader the proof of the following lemma:
Lemma 2.1. Let R0 be the following equivalence relation on E:
R0 ={(a, b)∈E×E :s(a) = s(b), r(a) =r(b)}
Then the map k :C∗(R0)→C∗(R) given by k(g)(xa, yb) =
g(a, b) if x=y 0 if x6=y identifies C∗(R0) to the commutant of C∗(R) in C∗(R).
Definition 2.1. A transition probability on the graph E is a function p:E →R∗+ such that for all v ∈V, P
s(c)=vp(c) = 1.
Proposition 2.2. Let X, V, E, V be as above and let p be a transition probability on E.
The map Q:C∗(R)→C∗(R) defined by Q(f)(x, y) =X
c
p(c)f(xc, yc),
where the sum is over all edges c∈E originating from the common range of x and y, is a faithful conditional expectation onto C∗(R).
Proof. This is a straighforward verification.
We are going to prove a converse to the proposition: namely all faithful conditional expectationsQ:M →M, whereM is a sub-C*-algebra of a finite dimensional C*-algebra M, are of that form. We first recall the notion of Cartan subalgebra which will be our main tool. It is an algebraic characterization of the canonical abelian subalgebra C(X) of the C*-algebra C∗(R) of an equivalence relation R onX as above.
Definition 2.2. An abelian subalgebra A of a von Neumann algebra M is called a Car- tan subalgebra if it is maximal self-adjoint, regular and there exists a faithful normal conditional expectation P :M →A. We then say that (M, A) is a Cartan pair.
Regularity means that the normalizer of A inM, which is defined here as NM(A) = {vpartial isometry ofM :vAv∗ ⊂A, v∗Av⊂A},
generates M as a von Neumann algebra. We recall the fact that the conditional expecta- tion P is unique.
The main result of [24] is that every Cartan pair (M, A) (if one assumes thatM acts on a separable Hilbert space) is of the form (W∗(R, τ), L∞(X, µ)) whereRis a countable Borel equivalence relation on a standard measured space (X, µ), where µ is a quasi-invariant measure and τ ∈Z2(R,T) is a Borel twist. When M is finite dimensional, the result of [24] is elementary: we let X be the spectrum of A and V be the spectrum of the centre Z(M) of M. The inclusion Z(M) ⊂ A gives a surjective map r : X → V. We let R be the equivalence relation admitting r as quotient map. Each x ∈ X corresponds to a minimal projection e(x) in A; e(x) and e(y) are equivalent if and only if (x, y)∈ R. We choose a matrix unit (e(x, y))(x,y)∈R such that for all x∈ X, e(x, x) = e(x). This matrix unit defines an isomorphism M → C∗(R) sending A to C(X). Thus, when M is finite dimensional, the twist is trivial. However, it does not admit a canonical trivialization.
Note also that in a finite dimensional C*-algebra, the notions of Cartan subalgebra and of maximal abelian self-adjoint subalgebra agree. We shall need an easy lemma about extension of matrix units.
Lemma 2.3. Let (M, A) be a finite dimensional Cartan pair and let (X, R) be the corre- sponding equivalence relation. Then every partial matrix unit (e(x, y))(x,y)∈S in M, where S is a subequivalence relation of R, can be extended to a full matrix unit (e(x, y))(x,y)∈R. Proof. We fix an arbitrary full matrix unit (e(x, y),(x, y) ∈ R). There exists a function c : S → T, where T is the group of complex numbers of module 1, such that e(x, y) = c(x, y)e(x, y) for all (x, y) ∈ S. It is a cocycle. Every cocycle on S is trivial: there exists b : X → T such that c(x, y) = b(x)b(y) for all (x, y) ∈ S. Then, we define
e(x, y) =b(x)e(x, y)b(y) for all (x, y)∈R.
The following lemma is a complement to Lemma III.1.14 of [40].
Lemma 2.4. Given an inclusion M ⊂ M of finite dimensional C*-algebras, a faithful conditional expectation Q : M → M and a Cartan subalgebra A of M, there exists a Cartan subalgebra A of M such that
(i) A⊂A,
(ii) NM(A)⊂NM(A), and
(iii) Q◦P =P ◦Q0, whereP is the conditional expectation from M ontoA, P is the conditional expectation from M to A and Q0 is the restriction of Q to A.
Proof. We let (X, R) be the equivalence relation defined by the pair (M, A): X is the spectrum ofA,V is the spectrum of the centreZ(M) ofM andr:X →V is the quotient map. We choose a matrix unit (e(x, y))(x,y) ∈ R of M with e(x, x) = e(x) minimal projection corresponding to x. We choose a section σ for the map r : X → V. For each v ∈V, we set Mv =e(σ(v))M e(σ(v)). There exists a unique state ϕ
v of the algebra Mv such that Q(f) = ϕ
v(f)e(σ(v)) for all f ∈Mv. It is faithful because Q is faithful. Since
self-adjoint matrices are diagonalizable, there exists a Cartan subalgebra Av of Mv such thatϕ
v =ϕ
v◦Pv, wherePv is the conditional expectation ontoAv. Forx∈X, we define Ax=e(x, σ(r(x)))Ar(x)e(σ(r(x)), x)
Then A=⊕x∈XAx is a Cartan subalgebra ofM. It contains A because for allx∈X,Ax contains e(x) as its unit element. By constructione(x, y) belongs to the normalizer of A inM, hence NM(A)⊂NM(A). Let v ∈V and a∈Mv. Then
P ◦Q(a) = P(ϕ
v(a)e(σ(v))) =ϕ
v(a)e(σ(v)).
On the other hand, since Pv is the restriction of P toMv, Q◦P(a) =ϕ
v(Pv(a))e(σ(v)) =ϕ
v(a)e(σ(v)).
Thus, P ◦Q and Q◦P agree on Mv Suppose now that a belongs to e(x)M e(y), where (x, y)∈ R. We write a=e(x, v)ave(v, y) with av ∈ Mv. Since e(x, v) and e(v, x) belong toM, Q(a) =e(x, v)Q(av)e(v, y) and sincee(x, v) and e(v, x) belong to NM(A),
P ◦Q(a) =e(x, v)P ◦Q(av)e(v, y).
On the other hand, sincee(x, v) ande(v, x) belong also toNM(A),P(a) = e(x, v)P(av)e(v, y).
Hence
Q◦P(a) =e(x, v)Q◦P(av)e(v, y).
Therefore,P ◦Qand Q◦P agree on e(x)M e(y). We deduce that they agree on M. This implies that Q(A) = A and that we have the equality Q◦P = P ◦Q0 where Q0 is the restriction of Q toA.
Definition 2.3. Let Q : M → M be a conditional expectation and let A, A be Cartan subalgebras of M, M respectively. We say that (M, A)⊂(M , A)
(i) is a Cartan pairs inclusion if it satisfies the conditions (i) and (ii) of the lemma;
(ii) is compatible withQ if moreover the condition (iii) of the lemma is also satisfied.
Lemma 2.5. Let (M, A) ⊂ (M , A) be an inclusion of finite dimensional Cartan pairs.
Then the spectrum X of A is canonically identified to the fibered product X×V E, where X is the spectrum ofA, E is the spectrum of M0∩A, V is the spectrum ofZ(M) and the fibered product is relative to the maps r : X →V and s :E →V given by the inclusions Z(M)⊂M and Z(M)⊂M0∩A.
Proof. We let α be the action of NM(A) on X and α be the action of NM(A) on X. We let π:X →X and π :X →E be the surjections corresponding to the inclusions A⊂A andM0∩A⊂A. They satisfyr◦π=s◦q. Hence (π, q) mapsX into the fibered product X×V E. This map is injective: letx, y ∈X such thatπ(x) = π(y) and q(x) =q(y). The elements of M0∩Aare exactly the functions on X which are constant under the actionα of NM(A) on X. Therefore, the relationq(x) =q(y) implies the existence of u∈NM(A) such that y=αu(x). This implies that π(x) =π(y) =αu(π(x)), hence ue(x) =e(x) and y=x. The map is surjective. Let (x, c)∈X×E such that r(x) =s(c). Picky∈X such that q(y) = c. Since r(π(y)) = r(x), there exists u ∈ NM(A) such that x = αu(π(y)).
Then x=αu(y) does the job.
An equivalent statement of the lemma is that A is canonically identified to A⊗Z(M) (M0∩A).
Lemma 2.6. Let (M, A) ⊂ (M , A) be an inclusion of finite dimensional Cartan pairs.
The commutant of M in M is denoted by M0.
(i) If a belongs to M0, then e(xa)ae(yb) = 0 if x6=y.
(ii) M0∩A is a Cartan subalgebra of M0.
(iii) the equivalence relation induced on the spectrum E of M0 ∩A by the normalizer is
R0 ={(a, b)∈E×E :s(a) = s(b), r(a) =r(b)}
Proof. i) Assume thatf commutes withM. If x6=y,
e(xa)f e(yb) = e(xa)e(x)f e(yb) =e(xa)f e(x)e(yb) = 0.
ii) For c∈ E, we denote by (c) the corresponding projection in M0 ∩A. According to (i), (c) =P
r(x)=s(c)e(xc). Suppose that f ∈M0 commutes with the elements ofM0∩A.
Consider xa and xb with a6=b. Then
e(xa)f e(xb) = e(xa)(a)f e(yb) = e(xa)f (a)e(yb) = 0.
Thuse(xa)f e(yb) = 0 if xa6=yb, therefore f belongs to A.
(iii) Assume that (a)M0(b) 6= 0. Then according to (i), there exists x ∈ X such that (xa, xb)∈R. This implies that (a, b)∈R0. Conversely, if (a, b)∈R0, we pick x∈X such that r(x) = s(a) = s(b). We choose a partial isometry u ∈ M such that uu∗ = e(xa), u∗u=e(xb). Then P
e(y, x)ue(x, y), where (e(x, y))(x,y)∈R is a matrix unit forR and the sum is over the y’s such that r(y) = r(x) is a partial isometry in M0 with domain (b) and range (a).
Lemma 2.7. Let (M, A) ⊂ (M , A) be an inclusion of finite dimensional Cartan pairs and letQ:M →M be a faithful conditional expectation which satisfies the condition (iii) of Lemma 2.4. Then, with above notations, there exists a transition probability p on the graph E such that for all c∈E,
Q((c)) =p(c)e(s(c))
where (c) is the minimal projection in M0 ∩A corresponding to c ∈ E and e(v) is the minimal projection in Z(M) corresponding to v ∈V.
Proof. As earlier, we denote byQ0 the restriction of QtoA. We first check thatQ0((c)) belongs toZ(M): for a∈M,
aQ0((c)) =Q0(a(c)) =Q0((c)a) = Q0((c))a
Then, we observe thate(x)(c) = 0 ifr(x)6=s(c). Thereforee(v)Q0((c)) = 0 for allv ∈V distinct from s(c): Q0((c)) is proportional to e(s(c)). The constant of proportionality is non zero because Qis supposed to be faithful. The equality e(v) = P
s(c)=v(c) gives the equality P
s(c)=vp(c) = 1.
Theorem 2.8. Let Q :M → M be a faithful conditional expectation on a finite dimen- sional C*-algebra and let A be Cartan subalgebra of M. We let (X, R) be the associated equivalence relation. Then,
(i) there exists A Cartan subalgebra of M such that (M, A) ⊂ (M , A) is a Cartan pairs inclusion compatible with Q.
(ii) any isomorphism Φ : M → C∗(R) carrying A onto C0(X) can be extended to an an isomorphism Φ : M → C∗(R) carrying Q into the model expectation Qp : C∗(R) → C∗(R) constructed from the graph (V, E, V) of the inclusion, the spectrum X of A and the transition probability p of Lemma 2.7.
Proof. The first assertion is Lemma 2.4. We fix a Cartan subalgebraA satisfying (i). We recall that the spectrum X of A can be identified with the fibered product X×V E and that the spectrum of M0 ∩A can be identified with E. We also recall from Lemma 2.6 that (M0, M0∩A) is a Cartan pair defining the equivalence relation R0 on E. We pick a matrix unit ((a, b))(a,b)∈R0 for the Cartan pair (M0, M0 ∩A). Let Φ :M →C∗(R) be an isomorphism carryingAontoC0(X). There exists a unique matrix unit (e(x, y))(x,y)∈Rfor the Cartan pair (M, A) which is sent by Φ onto the canonical matrix unit of C∗(R). We define e(xa, yb) =e(x, y)(a, b) if r(x) =r(y) =s(a) =s(b) and r(a) =r(b). This defines a partial matrix unit on a subequivalence relation of R. According to Lemma 2.3, it can be completed into a full matrix unit (e(xa, yb))(xa,yb)∈R. The isomorphism Φ :M →C∗(R) defined by this matrix unit extends Φ and satisfies Φ◦Q=Qp◦Φ.
In particular, the theorem shows that our path model of a conditional expectation gives every faithful conditional expectation. We can recover from this theorem the main result of [11], namely every conditional expectation on a finite dimensional C*-algebra can be written as a pinching followed by slicing and averaging: one introduces an intermediate level V1 in the inclusion graph (V, E, V), whose vertices label the edges (thus V1 = E).
The graph (V, E, V) is then written as the concatenation of two graphs (V, E1, V1) and (V1, E2, V). In the first graph, the vertices of V1 receive a single edge. In the second graph, the vertices ofV1 emit a single edge. With the ingredientX1 =X and (V1, E2, V), our recipe gives the inclusion C∗(R1) ⊂C∗(R) where (xa, yb) ∈R1 if and only if a = b.
The transition probability p2 ≡ 1 gives the restriction map Q2 :C∗(R)→ C∗(R1) as its associated conditional expectation. It is a pinching: in other words, it is of the form
Q2(f) =X
c∈E
(c)f (c).
The conditional expectationQ1 :C∗(R1)→C∗(R) is an averaging: for every v ∈V Q1(f)(x, y) = X
s(c)=v
p(c)f(xc, yc) for r(x) = r(y) = v.
In [7], A. Connes uses a similar decomposition of an inclusion of type I von Neumann algebras to construct inclusions of Cartan pairs.
3. Random walks on discrete Bratteli diagrams.
We first recall the classical definition of a Bratteli diagram.
Definition 3.1. A Bratteli diagram is a directed graph (V, E) where the set of vertices V = `∞
n=0V(n) and the set of edges E = `∞
n=1E(n) are graded. For each n ≥ 1, s(E(n)) = V(n−1) andr(E(n)) =V(n), wheres(e) andr(e) are respectively the source and the range of the edgee.
We assume that each level of verticesV(n) is at most countable; we also assumes that each vertex emits finitely many but at least one edge and that each vertex of a leveln ≥1 receives finitely many but at least one edge.
Definition 3.2. Let (V, E) be a Bratteli diagram.
• A transition probability is a map p assigning to each vertex v ∈V a probability measurep(v) on the set of edgesEv =s−1(v) emanating from v. We shall view p as a map p: E → R such that for all v ∈ V, P
s(e)=vp(e) = 1. We shall denote bypn its restriction to E(n).
• An initial probability measure is a probability measure ν0 on the set of initial verticesV(0).
• A random walk is a pair (p, ν0), where p is a transition probability and ν0 is an initial probability measure.
We shall always assume thatp and ν0 have full support, in the sense that p(e)>0 for alle ∈E and µ0(v)>0 for allv ∈V(0).
A Bratteli diagram (V, E) defines an ´etale equivalence relation (X, R) called the tail equivalence relation of the diagram: X is the set of infinite paths x = e1e2. . . where en ∈ E(n) and r(en) = s(en+1). It is a locally compact Hausdorff totally disconnected space admitting the cylinders
Z(a) = {aen+1en+2. . .}
where a =a1a2. . . an is a finite path (we assume implicitly that ai ∈ E(i)), as a base of compact open subsets. Two infinite pathsx=e1e2. . .andy =f1f2. . .are tail equivalent if there exists n such that ei =fi for i > n. Its graph R is a locally compact Hausdorff totally disconnected space admitting the cylinders
Z(a, b) = {(aen+1en+2. . . , ben+1en+2. . .)}
where (a, b) is a pair of equivalent finite paths: this means that they have same length n and same ranger(a) =r(b), where we define r(a1a2. . . an) =r(an)∈V(n).
A random walk on a Bratteli diagram defines a measure on the path space X; it is a particular case of the well-known construction of Markov measures.
Proposition 3.1. Given a random walk (p, ν0) on a Bratteli diagram (V, E), there is a unique probability measure µonX , called the Markov measure of the random walk whose values on cylinder sets is given by
µ(Z(a)) =ν0(s(a))p(a) where, for the finite path a =a1a2. . . an,
p(a) =p1(a1)p2(a2). . . pn(an) and s(a) =s(a1).
As observed in [41, Section 3.2], Markov measures are quasi-invariant under the tail equivalence relation. Let us recall that a measure µ on X is quasi-invariant under the equivalence relationRif the measuresr∗µands∗µonRare equivalent, whereR
f d(r∗µ) = R P
yf(x, y)dµ(x) for f ∈Cc(R) and s∗µ is similarly defined. Then its Radon-Nikodym derivativeDµ=d(r∗µ)/d(s∗µ) is a cocycle, i.e. it satisfiesDµ(x, y)Dµ(y, z) =Dµ(x, z) for a.e. (x, y, z) ∈ R(2). Here is a way to construct cocycles on the tail equivalence relation R of a Bratteli diagram (V, E).
Definition 3.3. Let G be a group. A map D :R →G is called a quasi-product cocycle if there exists a map q : E → G, called a potential, such that for all pairs of equivalent finite paths (a, b) and all (az, bz)∈Z(a, b),D(az, bz) =q(a)q(b)−1 and where, as before, q(a1a2. . . an) = q(a1)q(a2). . . q(an).
Since a quasi-product cocycle is locally constant, it takes at most countably many values and it is continuous. The following result, which is a simple observation, is essential here.
Proposition 3.2. [41, Proposition 3.3]Let(p, ν0)be a random walk on a Bratteli diagram (V, E).
(i) The associated Markov measure µ is quasi-invariant under the tail equivalence relation R
(ii) Its Radon-Nikodym derivative Dµ is the quasi-product cocycle given by the poten- tial q= (qn) defined by the relation
νn−1(s(e))pn(e) = qn(e)νn(r(e)) for e ∈E(n).
where νn is the distribution of the random walk on V(n), defined inductively by νn(w) =P
r(e)=wpn(e)νn−1(s(e)) for w∈V(n).
Note that the Radon-Nikodym derivative depends only on the potential q. This po- tential qhas a simple probabilitstic interpretation: it is the cotransition probabilityof the random walk: let e be an edge in E(n) with range r(e) =w, then q(e) is the probability that the random walk passes through e given that it is at w at time n. As shown by the next example, different random walks may share the same cotransition probability.
In his recent papers [44, 45], A. Vershik also emphasizes the importance of cotransition probabilities in the asymptotic study of random walks on Bratteli diagrams.
Example 3.1. Random walks on the Pascal triangle. It is the time development of the simple random walk onZ. Here, the Bratteli diagram is (V, E) where
V(n) = {(n, k) :k = 0,1, . . . , n}; E(n) ={(n−1, k, ) :k = 0,1, . . . , n−1;∈ {0,1}}.
We consider the random walk defined by the transition probability pn(n−1, k) = (1−t)δ(n−1,k,0)+tδ(n−1,k,1)
where 0< t <1. Since V(0) has a single vertex, the initial measure ν0 is the point mass at this vertex. The infinite path can be written as the infinite product X =Q∞
n=1{0,1}.
Then, the Markov measure is the product measure µt = Q∞
n=1((1 −t)δ0 + tδ1). An elementary computation gives the cotransition probability
qn(n, k) = (1− k
n)δ(n−1,k,0)+ k
nδ(n−1,k−1,1)
It does not depend on t. For a finite path 12. . . n ending at (n, k =1+. . .+n), one has
q(12. . . n) = n
k −1
.
One deduces that the Radon-Nikodym of µt is D ≡ 1. In other words, the measures µt are invariant under the tail equivalence relation on (V, E). It is a well-known result.
It is also well-known (see for example [42, Example 4.2]) that these are the extremal invariant probability measures (one has to add µ0 and µ1 which we have excluded from our discussion).
We use now the construction given by Feldman and Moore in [24]: since (X, R, µ) is a countable standard measured equivalence relation, one can construct its von Neumann algebraM=W∗(X, R, µ) and its stateϕ=µ◦P, whereP is the expectation ofMonto A = L∞(X, µ), which is normal and faithful. By construction, M acts on the Hilbert spaceL2(R, s∗µ). This representation is standard. It is known that the modular operator
∆ of ϕis the operator of multiplication by Dµ and that the modular automorphismσtϕ is implemented by the operator of multiplication by Dµit. A. Connes has given the following characterization of the pairs (M, ϕ) arising from this construction.
Theorem 3.3. [7, Theorem 1] Letϕbe a faithful normal state on a von Neumann algebra M. Then the following conditions are equivalent:
(i) there exists an increasing sequence (Mn) of finite dimensional subalgebras stable under the automorphism group σ of ϕ whose union is weakly dense inM;
(ii) there exists a Bratteli diagram(V, E)and a random walk(p, ν0)on it such that the pair(M, ϕ)is isomorphic to (W∗(X, R, µ), µ◦P), where R is the tail equivalence relation on the infinite path spaceX of the diagram and µis the Markov measure of the random walk.
The original theorem of Connes is in terms of Krieger’s factors. It is an intermediate step to show that all hyperfinite type III0 factors are Krieger’s factors. We consider here von Neumann algebras arising from hyperfinite measured equivalence relations rather than Krieger’s factors. This makes the statement easier to prove.
Proof. (ii)⇒(i). We assume that M = W∗(X, R, µ) as above. We let Mn be the sub- algebra of M generated by the characteristic functions 1Z(a,b), where (a, b) is a pair of joining paths of length n. Since the Radon-Nikodym derivative Dµ is constant on the cylinder sets Z(a, b), An is stable under the automorphism group of ϕµ. Since Z(a, b) is the disjoint union of Z(ae, be)’s where e ∈ E(n+ 1) and s(e) = r(a) = r(b), we have the inclusion Mn ⊂ Mn+1. The elements of the union M∞ of the Mn’s are the locally constant functions with compact support. SinceM∞is dense inCc(R) with respect to the inductive limit topology, it is dense in the weak topology. Since Cc(R) is weakly dense in M, so isM∞.
(i)⇒(ii). Let (Mn)n∈N be as in (i). Without loss of generality, we may assume that M0 = C1. Since for all n, Mn is stable under σ, the modular automorphism of the restriction ϕn to Mn is the restriction σn of σ to Mn. Since for all n ≥ 1, Mn−1 is
invariant under σn, there exists a faithful expectation Qn−1,n : Mn → Mn−1 such that ϕn=ϕn−1◦Qn−1,n. We use inductively Theorem 2.8, to construct an increasing sequence (An) of abelian subalgebras such that for all n≥1, (Mn−1, An−1)⊂(Mn, An) is a Cartan pair inclusion compatible with the conditional expectation Qn−1,n. The construction is initialized by the only possible choice A0 = M0. Thus we obtain for each n ∈ N the spectrum Xn of An and for eachn ≥1 the graph (V(n−1), E(n), V(n)) of the inclusion Mn−1 ⊂Mn and the transition probability pn :E(n)→R∗+. From the same theorem, we obtain for allnan isomorphism Φn :Mn →C∗(Rn) sending AntoC(Xn), where (Xn, Rn) is the equivalence relation defined by (Mn, An), such that Φn extends Φn−1 and carrying the conditional expectation Qn−1,n :Mn →Mn−1 into the model conditional expectation Fpn : C∗(Rn) → C∗(Rn−1). Again, the construction is initialized by the only possible isomorphism Φ0 :M0 →C(X0). Since the conditional expectationsPn:Mn →Ansatisfy Qn−1,n◦Pn =Pn−1 ◦Qn−1,n, we have ϕn = νn◦Pn, where νn is the restriction of ϕn to An. The Bratteli diagram of the increasing sequence (Mn) of finite dimensional algebras is (V =`∞
n=0V(n), E =`∞
n=1E(n)). The transition probabilitiespn :E(n)→R∗+define a transition probability on E. The initial measure ν0 is the point mass at the unique point ofX0. This defines the random walk. We letX be its infinite path space, R be the tail equivalence and µ be the Markov measure of the random walk. The isomorphisms Φn : Mn → C∗(Rn) extend to an isomorphism Φ∞ : M∞ → C00(R), where M∞ is the union of the Mn’s andC00(R) is the∗-algebra of locally constant functions with compact support. This isomorphism carries the restriction of the state ϕ to the restriction of the state µ◦P, where P is the expectation of W∗(R) onto L∞(X, µ). Since both von Neumann algebras Mand W∗(X, R, µ) can be obtained from the GNS representation of these states, Φ∞ extends to a normal ∗-isomorphism Φ : M →W∗(X, R, µ) which sends the weak closure A of the union of the An’s to L∞(X, µ) andϕ toµ◦P.
Remark 3.1. States on AF-algebras constructed from a random walk on a Bratteli diagram are called quasi-product states in [19]. Do we have a characterization (besides condition (i) of the theorem) of the normal faithful states on a hyperfinite von Neumann algebra which can be described as quasi-product states? Necessarily, theses states are almost periodic (their modular operators are diagonalizable) and their centralizers contain a Cartan subalgebra. In part II of [7], A. Connes shows that any faithful semifinite normal weight on a hyperfinite factor of type III0 whose modular operator ∆ is diagonalizable and such that 1 is isolated in its spectrum and its point spectrum contained inQsatisfies condition (i) of the theorem (with Mn type I∞ rather than finite-dimensional).
4. Markov chains and Bratteli diagrams
In order to recover the general theory of time-dependent Markov chains (with discrete time), it is necessary to generalize the notion of Bratteli diagram. Indeed, the original definition is limited to Markov chains with at most countably many states. Generalized Bratteli diagrams have been considered before, mostly in the topological setting, and are part of the theory of topological graphs. Since we are considering objects of measure- theoretical nature, we choose the Borel setting. We assume implicitly that the Borel spaces are analytic.
Definition 4.1. We say that a directed graph (V, E) is aBorel graphif the sets of edges E and the set of verticesV are endowed with a Borel structure and the source and range maps are Borel. A Borel Bratteli diagramis a Bratteli diagram which is a Borel graph.
Before extending to Borel Bratteli diagrams Definition 3.2, we need to make precise our assumptions.
Definition 4.2. Let E, V be Borel spaces and let s :E → V be a Borel surjection. An s-system of probability measurespis a map assigning to eachv ∈V a probability measure pv ons−1(v). We say that thes-system p is
(i) Borel if for all bounded Borel function f on E, the map v →R
f dpv is Borel;
(ii) ν-measurable, where ν is a probability measure on V, if for all bounded Borel function f on E, the map v →R
f dpv is ν-measurable.
If the s-system is ν-measurable, we can form the probability measure µ = νp on E, defined by R
f d(νp) = R (R
f dpv)dν(v) for f bounded Borel function on E. In fact it suffices to havepv defined for a.e. v to defineνp. The measure ν is the images∗(νp) ofνp and µ=νp is the disintegration of µ along s. Conversely, given a probability measure µ on E, we can disintegrateµ along s: there exists a ν-measurable s-system of probability measures p, whereν =s∗(µ), such thatµ=νp. It is unique in the sense that if νp=νp0, then pv =p0v for ν a.e.v.
Notation. Consider the n-th floorV(n−1)←−s E(n)−→r V(n) of a Borel Bratteli diagram.
A probability measure µn on E(n) admits a disintegration µn = νn−1pn along s and a disintegration µn = νnqn along r, where νn−1 = s∗µn and νn = r∗µn. This establishes a bijection between pairs (νn−1, pn),whereνn−1 is a probability measure on V(n−1) andpn
is aνn−1-measurable system of probability measures alongs and pairs (νn, qn),whereνn is a probability measure on V(n) and qn is a νn-measurable system of probability measures along r , given by the equation νn−1pn =νnqn.
Definition 4.3. Let (V, E) be a Borel Bratteli diagram.
• Arandom walk on (V, E) is a sequence of probability measuresµn onE(n) which are compatible in the sense that for all n≥1, r∗µn=s∗µn+1.
• The measures νn =r∗µn are called the one-dimensional distributions of the ran- dom walk.
• The measureν0 =s∗µ1 is called theinitial distribution of the random walk.
• The sequence p = (pn) of s-systems, where µ=νn−1pn as above, is called the transition probability of the random walk.
• The sequence q = qn of r-systems, where µn = νnqn as above, is called the cotransition probability of the random walk.
Remark 4.1. Note that by construction, the sequence (νn) of one-dimensional distributions satisfies the relations νn−1 = s∗(νnqn) and νn = r∗(νn−1pn for all n ≥. We say that it is q-compatible and p-compatible respectively.
Proposition 4.1. We have two constructions of a random walk on a Borel Bratteli dia- gram (V, E).
(i) Let ν0 be a probability measure on V(0) and let p = (pn) be a sequence of νn−1- measurable systems of probability measures for s : E(n) → V(n−1), where the measures µn = νn−1pn and νn = r∗(νn−1pn) are constructed inductively. Then there exists a unique random on (V, E) admittingν0 as initial distribution and p as transition probability.
(ii) Let (νn) be a sequence of probability measures νn on V(n) and let q = (qn) be a sequence of νn-measurable systems of probability measures for r : E(n) → V(n) which is q-compatible in the sense that νn−1 =s∗(νnqn) for all n≥1. Then there exists a unique random on(V, E) admitting(νn)as one-dimensional distributions and q as cotransition probability.
Proof. This is clear. In the first case, we define inductively µn = νn−1pn. In the second
case, we define µn =νnqn.
We recall the construction of the Markov measure of a random walk (see [39, V-1]).
As earlier, we introduce the infinite path space X. We let Xn denote the space of paths e1. . . en of length n endowed with the product Borel structure. Then X = lim←−Xn is the projective limit with respect to the canonical projection Xn ← Xn+1. Given a random walk (p, ν0), one first construct by induction a probability measure µn on Xn such that Z
f dµ1 = Z
f(e1)dpv(e1)dν0(v), Z
f dµn= Z
f(e1. . . en)dpr(en−1)(en)dµn−1(e1. . . en−1) The sequence of measures (µn) is consistent. Therefore, there exists a unique probability measureµonX whose image in Xn isµn. Note that the one-dimensional distribution νn onV(n) is the image of µn by the range map r:Xn →V(n). It is also the image of µby the map rn :X →V(n) such that rn(e1e2. . .) = r(en).
It remains to characterize the Markov measure µ on X in terms of the cotransition probability q. We have seen that in the framework of the previous section, the Markov measure µ is quasi-invariant under the tail equivalence relation and its Radon-Nikodym derivativeDis the quasi-product cocycle defined byq. We then say thatµis aD-measure.
The notion of quasi-product cocycle does not admit a straightforward generalization in the general framework. However, there exists (see [42, Proposition 3.7]) an equivalent definition of a D-measure (known in statistical mechanics as the Dobrushin-Lanford- Ruelle condition for Gibbs states) which can be easily extended. We let X|n be the space of infinite pathsen+1en+2. . .starting at level n. The sequence of quotient maps
X −π→1 X|1 −π→2 X|2 −π→3 . . .−→πn X|n−−−→π|n+1 . . .
defines the tail equivalence relationR onX: two infinite pathsxand yare tail equivalent if and only if there existn such thatπn◦. . . π2◦π1(x) =πn◦. . . π2◦π1(y). The cotransition probability q defines an inductive system of expectations
L∞(X, µ)−q→˜1 L∞(X1, µ|1)−q→˜2 . . .−q˜→n L∞(Xn, µ|n)−−→q˜n+1
˜
qn(f)(en+1en+2. . .) = Z
f(enen+1en+2. . .)dqs(en+1)(en).
Definition 4.4. Letqbe a cotransition probability on the Borel Bratteli diagram (V, E) A q-measureis a measure on the infinite path spaceXwhich factors through all expectations
˜
qn. . .q˜2q˜1.
Then we have the easy generalisation of Proposition 3.2:
Theorem 4.2. Let (V, E) be a Borel Bratteli diagram. Then the Markov measure of a random walk (p, ν0) is a q-measure, where q is the cotransition probability of the random walk.
Acknowledgements.
References
[1] C. Anantharaman-Delaroche and J. Renault: Amenable groupoids, Monographie de l’Enseignement Math´ematique No36, Gen`eve, 2000.
[2] N. Bourbaki: Topologie g´en´erale, chapitres 1`a 4, El´ements de Math´ematique, Diffusion C.C.L.S., Paris, 1971.
[3] O. Bratteli: Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc.,171(1971), 195–234.
[4] T. Bousch: La condition de Walters, Janvier 2000
[5] G. Brown and A. H. Dooley: Odometer actions on G-measures, Ergodic Theory Dyn. System, 11 (1991), 279–307.
[6] D. Capocaccia: A definition of Gibbs state for a compact set withZν action, Commun. Math. Phys., 48(1976), 85–88.
[7] A.-Connes: On hyperfinite factors of type III0 and Krieger’s factors, J. Funct. Anal., 18 (1975), 318–327.
[8] A. Connes and E. J. Woods: Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks, Pacific J. Math. 137(1989), no 2, 225–243.
[9] J. Cuntz: A class of C∗-algebras and topological Markov chains II: reducible chains and the Ext- functor forC∗-algebras, Inventiones Math.,63(1981), 25–40.
[10] J. Cuntz and W. Krieger: A class ofC∗-algebras and topological Markov chains, Inventiones Math., 56(1980), 251–268.
[11] C. Davis: Various averaging operations on subalgebras, Illinois J. Math.,3(1959), 538–553.
[12] V. Deaconu: Groupoids associated with endomorphisms, Trans. Amer. Math. Soc.,347(1995), 1779–
1786.
[13] E. Dynkin: The exit space of a Markov process, Russian Math. Surveys24, No. 4, (1969), 89–152.
[14] E. Effros: Dimensions and C∗-algebras, CBMS Regional Conf. Series in Math., 46, Amer. Math.
Soc., 1981.
[15] G. Elliott: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra,38(1976), 29–44.
[16] M. Enomoto, M. Fujii and Y. Watatani: KMS states for gauge action on OA, Math. Japon., 29 (1984), 607-619.
[17] D. E. Evans: OnOn, Publ. RIMS Kyoto Univ.16(1980), 915-927.
[18] D. E. Evans: TheC∗-algebras of topological Markov chains, Tokyo Metropolitan University Lecture Notes (1983).
[19] D. E. Evans: Quasi-product states on C∗-algebras, Lecture Notes in Mathematics, No. 1132, Springer- Verlag, Berlin-New York, 1985, 129-151.
[20] R. Exel: KMS states for generalized gauge actions on Cuntz-Krieger algebras, preprint, (2001).
[21] R. Exel and M. Laca: Cuntz-Krieger algebras for infinite matrices, J. reine angew. Mth. (Crelle) 512(1999), 119–172.
[22] R. Exel and M. Laca: Partial Dynamical Systems and the KMS Condition, preprint, (2000).
[23] T. Fack and O. Mar´echal: Sur la classification des sym´etrie des C∗-alg`ebres UHF, Canadian Journal of Math.
[24] J. Feldman and C. Moore: Ergodic equivalence relations, cohomologies, von Neumann algebras, I and II, Trans. Amer. Math. Soc.,234(1977), 289-359.
[25] W. Feller: An introduction to probability theory and its applications, vol. II, John Wiley, N.Y.
[26] A. H. Fan: A proof of the Ruelle theorem, Reviews Math. Phys.7, no. 8 (1995), 1241–1247.
[27] A. H. Fan and Y. P. Jiang: On Ruelle-Perron-Frobenius Operators. I. Ruelle theorem, Commun.
Math. Phys.223(2001), 125–141.
[28] H.-O. Georgii: Gibbs measures and phase transitions, de Gruyter Studies in Mathematics, 1988.
[29] T. Giordano, I. Putnam and C. Skau: Affable equivalence relations and orbit structure of Cantor dynamical systems, preprint 2002.
[30] K. R. Goodearl: Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs,20, Amer. Math. Soc., 1986.
[31] G. Keller: Equilibrium States in Ergodic Theory, London Mathematical Society Student Texts42, Cambridge University Press (1998).
[32] S. Kerov: The boundary of Young lattice and random Young tableaux, DIMACS Ser. Discrete Math.
Theoret. Comput. Sci.24, Amer. Math. Soc., (1996), 133–158.
[33] D. Kerr and C. Pinzari: Noncommutative pressure and the variational principle in Cuntz-Krieger type C*-algebras, J. Funct. Anal.,188(2002), 156-215.
[34] W. Krieger: On dimension functions and topological Markov chains, Inventiones Math.,56 (1980), 239–250.
[35] A. Kumjian: On localizations and simple C*-algebras, Pacific J. Math. 112(1984), 141–192.
[36] A. Kumjian and J. Renault: KMS states on C∗-algebras associated to expansive maps, in preparation.
[37] D. Lind and B. Marcus: An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995.
[38] P. Muhly, D. Williams: Continuous trace groupoid C*-algebras, Math. Scand.66(1990), 231–241.
[39] J. Neveu: Bases math´ematiques du calcul des probabilit´es, Masson & Cie, Paris, 1964.
[40] J. Renault: A groupoid approach toC∗-algebras, Lecture Notes in Mathematics, Vol.793Springer- Verlag Berlin, Heidelberg, New York (1980).
[41] J. Renault: AF equivalence relations and their cocycles, Operator Algebras and Mathematical Physics, Conference Proceedings, Constanza 2001, the Theta Foundation (2003), 365-377.
[42] J. Renault: The Radon-Nikodym problem for approximately proper equivalence relations, Ergodic Theory Dyn. System,25(2005), 1643–1672.
[43] A. T¨or¨ok: AF-algebras with unique trace, Acta. Sci. Math. (Szeged)55, No 1-2 (1991), 129–139.
[44] A. Vershik Equipped graded graphs, projective limits of simplices, and their boundaries, Zapiski Nauchn. Semin. POMI432, (2015), 83–104, English translation to appear in J. Math. Sci.
[45] A. VershikAsymptotic theory of path spaces of graded graphs and its applications, Japanese J. Math.
11, No. 2, (2016), 151–218 .
[46] A. Vershik and S. Kerov: Asymptotic character theory of the symmetric group, Funct. Analysis and its Applications15, (1981), 25–36.
[47] P. Walters: Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc.,236, (1978) , 121–153.
[48] P. Walters,An introduction to ergodic theory, Graduate Texts in Mathematics,79, Springer-Verlag, New York-Berlin, 1982.
[49] P. Walters: Convergence of the Ruelle operator for a function satisfying Bowen’s condition, Trans.
Amer. Math. Soc.,353, No 1 (2000) , 327–347.
[50] A. Wassermann: Automorphic actions of compact groups on operator algebras, Ph.D thesis, U. of Pennsylvania (1981).
[51] M. Zinsmeister: Formalisme thermodynamique et syst`emes dynamiques holomorphes, Panoramas et Synth`eses,4, Soci´et´e Math´ematique de France, 1996.
D´epartment de Math´ematiques, Universit´e d’Orl´eans, 45067 Orl´eans, France E-mail address: [email protected]
